Petri net model using Petri.jl

Micah Halter (@mehalter), 2020-07-14

Introduction

This implementation considers the SIR model as a Petri net, using Petri.jl, which is then used to generate ODE, SDE, and jump process models.

Libraries

using Petri
using LabelledArrays
using OrdinaryDiffEq
using StochasticDiffEq
using DiffEqJump
using Random
using Plots

Transitions

The Petri model is specified using a vector of the model states (as symbols), and a labelled vector of the transition rates; in this case, inf (infection) and rec (recovery). Each transition is a tuple of labeled vectors with inputs and outputs.

sir = Petri.Model([:S,:I,:R],LVector(
                                inf=(LVector(S=1,I=1), LVector(I=2)),
                                rec=(LVector(I=1),     LVector(R=1))))

Petri.Model{Array{Symbol,1},LabelledArrays.LArray{Tuple{LabelledArrays.LArr
ay{Int64,1,Array{Int64,1},Syms} where Syms,LabelledArrays.LArray{Int64,1,Ar
ray{Int64,1},Syms} where Syms},1,Array{Tuple{LabelledArrays.LArray{Int64,1,
Array{Int64,1},Syms} where Syms,LabelledArrays.LArray{Int64,1,Array{Int64,1
},Syms} where Syms},1},(:inf, :rec)}}([:S, :I, :R], 2-element LabelledArray
s.LArray{Tuple{LabelledArrays.LArray{Int64,1,Array{Int64,1},Syms} where Sym
s,LabelledArrays.LArray{Int64,1,Array{Int64,1},Syms} where Syms},1,Array{Tu
ple{LabelledArrays.LArray{Int64,1,Array{Int64,1},Syms} where Syms,LabelledA
rrays.LArray{Int64,1,Array{Int64,1},Syms} where Syms},1},(:inf, :rec)}:
 :inf => (2-element LabelledArrays.LArray{Int64,1,Array{Int64,1},(:S, :I)}:
 :S => 1
 :I => 1, 1-element LabelledArrays.LArray{Int64,1,Array{Int64,1},(:I,)}:
 :I => 2)
 :rec => (1-element LabelledArrays.LArray{Int64,1,Array{Int64,1},(:I,)}:
 :I => 1, 1-element LabelledArrays.LArray{Int64,1,Array{Int64,1},(:R,)}:
 :R => 1))

Using Graphviz, a graph showing the states and transitions can also be generated from the Petri net.

Graph(sir)

Time domain

tmax = 40.0
tspan = (0.0,tmax);

(0.0, 40.0)

Initial conditions

u0 = LVector(S=990.0, I=10.0, R=0.0)

3-element LabelledArrays.LArray{Float64,1,Array{Float64,1},(:S, :I, :R)}:
 :S => 990.0
 :I => 10.0
 :R => 0.0

Parameter values

p = LVector(inf=0.5/sum(u0), rec=0.25);

2-element LabelledArrays.LArray{Float64,1,Array{Float64,1},(:inf, :rec)}:
 :inf => 0.0005
 :rec => 0.25

Random number seed

We set a random number seed for reproducibility.

Random.seed!(1234);

Random.MersenneTwister(UInt32[0x000004d2], Random.DSFMT.DSFMT_state(Int32[-
1393240018, 1073611148, 45497681, 1072875908, 436273599, 1073674613, -20437
16458, 1073445557, -254908435, 1072827086  …  -599655111, 1073144102, 36765
5457, 1072985259, -1278750689, 1018350124, -597141475, 249849711, 382, 0]),
 [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 
0.0, 0.0, 0.0, 0.0, 0.0, 0.0], UInt128[0x00000000000000000000000000000000, 
0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x0
0000000000000000000000000000000, 0x00000000000000000000000000000000, 0x0000
0000000000000000000000000000, 0x00000000000000000000000000000000, 0x0000000
0000000000000000000000000, 0x00000000000000000000000000000000, 0x0000000000
0000000000000000000000  …  0x00000000000000000000000000000000, 0x0000000000
0000000000000000000000, 0x00000000000000000000000000000000, 0x0000000000000
0000000000000000000, 0x00000000000000000000000000000000, 0x0000000000000000
0000000000000000, 0x00000000000000000000000000000000, 0x0000000000000000000
0000000000000, 0x00000000000000000000000000000000, 0x0000000000000000000000
0000000000], 1002, 0)

Generating and running models

As ODEs

prob_ode = ODEProblem(sir,u0,tspan,p)
sol_ode = solve(prob_ode, Tsit5());
plot(sol_ode)

As SDEs

prob_sde,cb = SDEProblem(sir,u0,tspan,p)
sol_sde = solve(prob_sde,LambaEM(),callback=cb);
plot(sol_sde)

As jump process

prob_jump = JumpProblem(sir, u0, tspan, p)
sol_jump = solve(prob_jump,SSAStepper());
plot(sol_jump)

Julia Version 1.4.2
Commit 44fa15b150* (2020-05-23 18:35 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Core(TM) i7-8850H CPU @ 2.60GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
Environment:
  JULIA_HOME = /home/micah/.local/share/julia
  JULIA_LOAD_PATH = :
  JULIA_DEPOT_PATH = /home/micah/Documents/git/sir-julia/env/.julia
  SPACESHIP_JULIA_SYMBOL = ∴

Status `~/Documents/git/sir-julia/Project.toml`
[46ada45e-f475-11e8-01d0-f70cc89e6671] Agents 3.2.1
[4f99eebe-17bf-4e98-b6a1-2c4f205a959b] AlgebraicPetri 0.3.1
[b19378d9-d87a-599a-927f-45f220a2c452] ArrayFire 1.0.6
[c52e3926-4ff0-5f6e-af25-54175e0327b1] Atom 0.12.16
[6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf] BenchmarkTools 0.5.0
[be33ccc6-a3ff-5ff2-a52e-74243cff1e17] CUDAnative 3.2.0
[134e5e36-593f-5add-ad60-77f754baafbe] Catlab 0.7.1
[3a865a2d-5b23-5a0f-bc46-62713ec82fae] CuArrays 2.2.2
[717857b8-e6f2-59f4-9121-6e50c889abd2] DSP 0.6.7
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[a93c6f00-e57d-5684-b7b6-d8193f3e46c0] DataFrames 0.21.4
[459566f4-90b8-5000-8ac3-15dfb0a30def] DiffEqCallbacks 2.13.3
[aae7a2af-3d4f-5e19-a356-7da93b79d9d0] DiffEqFlux 1.17.0
[c894b116-72e5-5b58-be3c-e6d8d4ac2b12] DiffEqJump 6.9.3
[41bf760c-e81c-5289-8e54-58b1f1f8abe2] DiffEqSensitivity 6.23.0
[6d1b261a-3be8-11e9-3f2f-0b112a9a8436] DiffEqTutorials 0.1.0
[0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.15.0
[31c24e10-a181-5473-b8eb-7969acd0382f] Distributions 0.23.4
[634d3b9d-ee7a-5ddf-bec9-22491ea816e1] DrWatson 1.14.3
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[0c68f7d7-f131-5f86-a1c3-88cf8149b2d7] GPUArrays 3.4.1
[28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71] GR 0.50.1
[523d8e89-b243-5607-941c-87d699ea6713] Gillespie 0.1.0
[7073ff75-c697-5162-941a-fcdaad2a7d2a] IJulia 1.21.2
[e5e0dc1b-0480-54bc-9374-aad01c23163d] Juno 0.8.2
[2ee39098-c373-598a-b85f-a56591580800] LabelledArrays 1.3.0
[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.13.0
[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.22.0
[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
[4259d249-1051-49fa-8328-3f8ab9391c33] Petri 1.1.0
[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.4
[e6cf234a-135c-5ec9-84dd-332b85af5143] RandomNumbers 1.4.0
[c5292f4c-5179-55e1-98c5-05642aab7184] ResumableFunctions 0.5.1
[428bdadb-6287-5aa5-874b-9969638295fd] SimJulia 0.8.0
[05bca326-078c-5bf0-a5bf-ce7c7982d7fd] SimpleDiffEq 1.1.0
[2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91] StatsBase 0.33.0
[f3b207a7-027a-5e70-b257-86293d7955fd] StatsPlots 0.14.6
[789caeaf-c7a9-5a7d-9973-96adeb23e2a0] StochasticDiffEq 6.24.0
[44d3d7a6-8a23-5bf8-98c5-b353f8df5ec9] Weave 0.10.2
[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
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